With properly chosen parameters, as in the \(q\) -TAZRP to be described below, the 1-dimensional ZRP model is integrable, and is solvable by the Bethe ansatz [1]}, [2]}, [3]}. The application of the Bethe ansatz to find the explicit transition probabilities in 1-dimensional particle models has come along two decades since [4]} and [5]}, and it has yielded the explicit transition probability formulas for the asymmetric simple exclusion process (ASEP) in Tracy and Widom's seminal work [6]}. These exact formulas enabled (after further manipulation) Tracy and Widom to calculate the asymptotic distribution of one particle as the number of particles approaches infinity with special initial conditions [7]}, and greatly improve our understanding of the KPZ universality [8]}, [9]}. Since then, different methods have been developed and exact formulas for the transition probabilities have been studied for various integrable 1-dimensional particle models [10]}, including the \(q\) -deformed totally asymmetric simple exclusion process (\(q\) -TASEP), which is an example of the general framework of Macdonald processes [11]} and has duality with \(q\) -TAZRP [1]}.
